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प्रश्न
Using Biot − Savart’s law, derive the expression for the magnetic field in the vector form at a point on the axis of a circular current loop?
उत्तर
Magnetic field on the axis of a circular current loop
I →Current in the loop, R →Radii of the loop, X-axis →Axis of the loop, x →Distance of OP
dl →Conducting element of the loop
According to Biot-Savart’s law, the magnetic field at P is `dB =(μ_0I|dI xx r)/(4pir^3)`
r2 = x2+ R2
| dl × r | = r dl (∵ they are perpendicular)
`thereforedB = (μ_0)/(4pi) (IdI)/((x^2 +R^2))`
dB has two components − dBx and dB⊥. dB⊥ is cancelled out and only the x-component remains.
∴ dBx= dBcosθ
`Cos theta = R/((x^2 + R^2)^(1/2)`
`therefore dB_x =(μ_0IdI)/(4pi) R/(x^2 +R^2)^(3/2)`
Summation of dl over the loop is given by 2πR.
`B =B_xhati =(μ_0IR^2)/(2(x^2 + R^2)^(3/2 ) )hati`
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